You don’t need to stop the Mattermost service to run an export with either bin/mattermost export bulk or mmctl export create. Both commands are designed to run against a live instance.
Stopping the service is only necessary if you want to guarantee that no new messages or attachments are created during the export (to ensure a perfectly consistent snapshot). In most environments, the small window of changes that might occur during export is acceptable.
Documentation reference: Bulk Export Tool. It notes that exports run against a running server, though it doesn’t explicitly say to stop the service.
So in practice:
For normal exports → safe to run without downtime.
For a fully consistent snapshot → briefly stop Mattermost, run the export, then start it again.